Williams, back after an injury lay-off, demolished the dangerous Belinda Bencic 6-4, 6-3, while Nadal was also impressive in his 6-3, 6-4, 6-4 win over Florian Mayer.
It contrasts with the laboured progress of Angelique Kerber and Andy Murray a day earlier, despite temperatures rising to 37 Celsius (98.6 Fahrenheit) on day two.
Men's defending champion Novak Djokovic is due on court in thecooler, evening session when he will play his first-round match against Spanish left-hander Fernando Verdasco.
Williams has barely played since her shock defeat in the US Open semi-finals in September and she made, by one count, 88 unforced errors in her second-round loss to Madison Brengle at this month's Auckland Classic.
But she came through a tight first set against Bencic before dominating the second set in what turned out to be a convincing victory on centre court.
The 35-year-old American, with a record 23rd major title in her sights, was watched from courtside by her new fiance, Reddit co-founder Alexis Ohanian.
"It was one of the toughest first-round matches I've ever played," said Williams, who had a 1-1 head-to-head with Bencic, 19, before Tuesday's encounter.
"Right now I've got nothing to lose. Every match, every tournament I'm playing for fun," she added.